Question

consider the function f(x) graphed below. for this function, are the following nonzero quantities positive or negative? f(0.5) is ? f(0.5) is ? f(0.5) is ?

Explanation:

Step1: Determine $f(0.5)$ value

Observe the graph at $x = 0.5$. The function value $f(0.5)$ is above the $x -$axis, so $f(0.5)>0$.

Step2: Determine $f'(0.5)$ sign

The derivative $f'(x)$ represents the slope of the tangent line. At $x = 0.5$, the tangent - line to the graph has a positive slope (the function is increasing), so $f'(0.5)>0$.

Step3: Determine $f''(0.5)$ sign

The second - derivative $f''(x)$ represents the concavity. At $x = 0.5$, the graph is concave down (the curve bends downwards), so $f''(0.5)<0$.

Answer:

$f(0.5)$ is positive; $f'(0.5)$ is positive; $f''(0.5)$ is negative.